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CSCE 1020.002
Binary and Hexadecimal Numbers
Binary Numbers
Computers store and process data in terms of binary numbers.
Binary numbers consist of only the digits 1 and 0.
It is important for Computer Scientists and Computer Engineers to understand how binary numbers work.
2Note: “Binary Numbers” are also referred to as “Base 2” numbers.
Review of Placeholders
You probably learned about placeholders in the 2nd or 3rd grade. For example:
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31251’s place10’s place100’s place1000’s place
So this number represents • 3 thousands• 1 hundred• 2 tens• 5 ones
Mathematically, this is
(3 x 1000) + (1 x 100) + (2 x 10) + (5 x 1)= 3000 + 100 + 20 + 5 = 3125
But why are the placeholders 1, 10, 100, 1000, and so on?
More on Placeholders
The numbers commonly used by most people are in Base 10.
The Base of a number determines the values of its placeholders.
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312510
100 place101 place102 place103 place
To avoid ambiguity, we often write the base of a number as a subscript.
Binary Numbers - Example
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20 place21 place22 place23 place
10102
This subscript denotes that this number is in Base 2 or “Binary”.
1’s place2’s place4’s place8’s place
Binary Numbers - Example
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10102
1’s place2’s place4’s place8’s place
So this number represents • 1 eight• 0 fours• 1 two• 0 ones
Mathematically, this is
(1 x 8) + (0 x 4) + (1 x 2) + (0 x 1)= 8 + 0 + 2 + 0 = 1010
Which Digits Are Available in which Bases
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Base 10 0 1 2 3 4 5 6 7 8 910
Base 2 0 110
10 d
igits
2 di
gits
Base 16 0 1 2 3 4 5 6 7 8 9 A B C D E F10
16 d
igits
Note: Base 16 is also called “Hexadecimal” or “Hex”.
Base 16Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Add Placeholder
Add Placeholder
Add Placeholder
Hexadecimal Numbers - Example
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160 place161 place162 place
3AB16
This subscript denotes that this number is in Base 16 or “Hexadecimal” or “Hex”.
1’s place16’s place256’s place
Note:162 = 256
Hexadecimal Numbers - Example
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3AB16
1’s place16’s place256’s place
So this number represents • 3 two-hundred fifty-sixes• 10 sixteens• 11 ones
Base 16Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Mathematically, this is
(3 x 256) + (10 x 16) + (11 x 1)= 768 + 160 + 11 = 93910
Why Hexadecimal Is Important
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What is the largest number you can represent using four binary digits?
_ _ _ _2
1 1 1 1
23 22 21 20
8 4 2 1
====
8 + 4 + 2 + 1 = 1510
… the smallest number?_ _ _ _
20 0 0 0
23 22 21 20
0 + 0 + 0 + 0 = 010
What is the largest number you can represent using a single hexadecimal digit?
Base 16Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
_16
F = 1510
… the smallest number?
_16
0 = 010 Note: You can represent the same range of values with a single hexadecimal digit that you can represent using four binary digits!
Why Hexadecimal Is ImportantContinued
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It can take a lot of digits to represent numbers in binary.
Example:5179410 = 11001010010100102
Long strings of digits can be difficult to work with or look at.
Also, being only 1’s and 0’s, it becomes easy to insert or delete a digit when copying by hand.
Hexadecimal numbers can be used to abbreviate binary numbers.
Starting at the least significant digit, split your binary number into groups of four digits.
Convert each group of four binary digits to a single hex digit.
Converting Binary Numbers to Hex
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Recall the example binary number from the previous slide:11001010010100102
1100 1010 0101 00102
First, split the binary number into groups of four digits, starting with the least significant digit.
Next, convert each group of four binary digits to a single hex digit.
C A 5 2
Base 16Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
Put the single hex digits together in the order in which they were found, and you’re done!
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In many situations, instead of using a subscript to denote that a number is in hexadecimal, a “0x” is appended to the front of the number.
Look! Hexadecimal Numbers!
Windows“Blue Screen of Death”
Converting Decimal to Binary
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Example:We want to convert 12510 to binary.
125 / 2 = 62 R 1 62 / 2 = 31 R 0 31 / 2 = 15 R 1 15 / 2 = 7 R 1 7 / 2 = 3 R 1 3 / 2 = 1 R 1 1 / 2 = 0 R 1
12510 = 11111012
Converting Decimal to Hex
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Example:We want to convert 12510 to hex.
125 / 16 = 7 R 13 7 / 16 = 0 R 7
12510 = 7D16
Base 16Cheat Sheet
A16 = 1010
B16 = 1110
C16 = 1210
D16 = 1310
E16 = 1410
F16 = 1510
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